Convergence conditions for Newton-type methods applied to complementarity systems with nonisolated solutions

Abstract

Abstract. We consider a class of Newton-type methods for constrained systems of equations that involve complementarity conditions. In particular, at issue are the constrained Levenberg-Marquardt method and the recently introduced Linear-Programming-Newton method, designed for the difficult case when solutions need not be isolated, and the equation mapping need not be differentiable at the solutions. We show that the only structural assumption needed for rapid local convergence of those algorithms is the piecewise error bound, i.e., a local error bound holding for the branches of the solution set resulting from partitions of the bi-active complementarity indices. The latter error bound is implied by various piecewise constraint qualifications, including some relatively weak ones. As further applications of our results, we consider Karush-Kuhn-Tucker systems arising from optimization or variational problems, and from generalized Nash equilibrium problems. In the first case, we show convergence under the assumption that the dual part of the solution is a noncritical Lagrange multiplier, and in the second case convergence follows under a certain relaxed constant rank condition. In both cases, this improves on results previously available for the corresponding problems.

On the constrained error bound condition and the projected Levenberg–Marquardt method

“…Therefore, since s ∈ \ X * implies dist[s, O H ] > 0, Lemma 3.2 yields Thus, in particular, the fractions within assertions (a) and (b) are well-defined. Clearly, using (15) and Lemma 3.4 (a), we have…”

“…Moreover, it is well known that error bound conditions weaken classical regularity conditions and play a decisive role in the design and local convergence analysis of other Newton-type methods as well. [9][10][11][12][13][14][15][16][17][18] The citations related to the use of error bound conditions can be a selection only and, further, do not cover all fields where error bounds are of importance. Note that error bound conditions, sometimes also linked to notions of calmness and upper Lipschitz continuity, might be fulfilled at nonisolated solutions, differently from what happens in the classical case, i.e.…”

A Globally Convergent LP-Newton Method

“…follow, provided that z k is close enough toz. Now, local convergence results of Algorithm 1 can be derived using those for the original LP-Newton method in [8]; see also [13].…”

“…An important class of problems that gives rise to piecewise smooth equations is systems that involve complementarity conditions. For example, consider the problem of finding a point z ∈ R n such that Convergence conditions for the associated LP-Newton method are analyzed in detail in [13]. The importance of this specific choice of Ω is discussed in [8,13].…”

“…We emphasize that the LP-Newton method has very strong local convergence properties. In particular, the assumptions for local quadratic convergence to a solution of (1.1) neither imply differentiability of F nor require the solutions to be isolated; see [8,13]. This combination of difficulties is very challenging, and other methods require stronger assumptions; see the discussion in [8].…”